(ed.). Gravitational waves (IOP, 2001)(422s).

262 Infinite-dimensional symmetries in gravityThe dependence of this equation on ρ is all that remains of three-dimensionalgravity. Equation (14.126) reduces to the Ernst equation for G = SL(2, R), butwe will return to this later. For the moment, note only that this equation works forthe σ -model degrees of freedom, namely on .The remaining equations, which follow from higher dimensions, are theequations for the dilaton ρ and for the conformal factor λ. ρ is a free field intwo dimensions which can be solved for in terms of two arbitrary functions (leftmoversand right-movers)£ρ = 0 ⇒ ρ(x) = ρ + (x + ) + ρ − (x − ). (14.127)The equations of motion for the conformal factor in light-cone coordinatesρ −1 ∂ ± ρλ −1 ∂ ± λ = 1 2 Tr(P ± P ± ) + 1 2 ρ−1 ∂ 2 ± ρ (14.128)can be written as∂ ± ρ∂ ± ˆσ = 1 2 ρ Tr P ± P ± (14.129)where the second term on the right-hand side of (14.128) has been reabsorbed intothe Liouville scalar ˆσ = λ(∂ + ρ) − 1 2 (∂ − ρ) − 1 2 . Note that this equation determinesλ only up to a constant factor. Observe also that this equation has no analoguein flat space theories, and this, together with the presence of ρ, makes a greatdifference. For instance, we cannot simply put ρ = constant, for this wouldimply the vanishing of the right-hand side of (14.129), which by the positivity ofthe Killing metric on the subalgebra K would imply P ± = 0 and leave us onlywith the trivial solution v = constant (modulo H gauge transformations).Now specializing to general relativity, i.e. G/H = SL(2, R)/SO(2) cosetspace. As anticipated before, we start from the equation of motion (14.126),employ the triangular gauge in the Ehlers form, so to have explicit expressionsfor P µ and Q ν . After a little algebra we have again the Ernst equation∂ µ (ρ∂ µ ) = ρ∂ µ ∂ µ (14.130)in terms of the complex potential = + iB. Solving Einstein’s equationsis now simply a matter of choosing the appropriate ρ(x), finding a solution ofthe nonlinear partial differential equation (14.130) and finally determining theconformal factor λ by integration of (14.129). For the colliding plane wavesolutions, that will be recovered in the next sections, one distinguishes waves withcollinear polarization, where B = 0 and waves with non-collinear polarization.For collinearly polarized waves, the nonlinear Ernst equation can be reduced to alinear partial differential equation through the replacement = exp ψ.So, for collinearly polarized waves, with B = 0, the four-bein is⎛λ −1/2 ⎞0 0 0E A ⎜ 0 λM =−1/2 0 0 ⎟⎝0 0 ρ −1/2 ⎠ (14.131)00 0 0 1/2